Based on $G$-hulls and $G$-kernels under the meaning of $G$-methods on sets, we introduce the concepts of $G$-hull-closed sets, $G$-kernel-open sets, $G$-kernel-neighborhoods and ~$G$-kernel-derived sets, discuss some related properties. In particular, we define pointwise $G$-methods, prove the consistency of $G$-closed sets and $G$-hull-closed sets, $G$-open sets and $G$-kernel-open sets, $G$-neighborhoods and $G$-kernel-neighborhoods, $G$-derived sets and~$G$-kernel-derived sets under this method, and enrich some results about $G$-closed sets, $G$-open sets, $G$-interiors, $G$-neighborhoods and $G$-derived sets in sets. At the same time, we put forward some problems for further research.